# How Applicable Are Turbulence Assumptions Used in the Tidal Energy Industry?

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## Abstract

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## 1. Introduction

## 2. Review of Models

#### 2.1. Frequency

#### 2.1.1. Turbulence Spectrum

#### 2.1.2. Spatial Coherence

#### 2.1.3. Length Scales

#### 2.1.4. Mean Velocity Profile

#### 2.1.5. Turbulence Intensity

## 3. Literature Review

## 4. Methods

#### 4.1. Measurements

#### 4.2. Data Analysis

#### 4.2.1. Turbulence Spectrum

`pwelch`was used to compute the discrete Fourier transform of each 10-min time series using a fast Fourier transform (fft) algorithm. It allows specification for window size, type and overlap. By default, the time series is divided into the longest possible segments to obtain as close to, but not exceeding, eight segments, with 50% overlap, using a Hamming window. The modified periodograms are averaged to obtain the power spectral density (PSD) estimate. The default number of discrete Fourier transform (DFT) points is the greater of 256 or the next power of 2 greater than the length of the segments. In the PSD format, the area under the spectrum curve represents the variance for the data record.

#### 4.2.2. Coherence

`mscohere`to calculate coherence for each beam. This function finds the magnitude-squared coherence estimate for two input signals. ADCP’s do not provide a direct measurement of the streamwise velocity component, so we applied the coordinate transform to instantaneous beam velocities to obtain the streamwise component (making the assumption of homogeneity across the beam spread). These values were then used in the coherence calculation. Due to the uncertainty relating to the homogeneity assumption, we also calculated coherence using the raw beam velocities for comparison.

#### 4.2.3. Length Scales

#### 4.2.4. Shear Profile

#### 4.2.5. Turbulence Intensity

#### 4.2.6. ADCP Alignment Error

## 5. Results

#### 5.1. Length Scales

#### 5.2. Turbulence Spectrum

#### 5.3. Coherence

#### 5.4. Shear Profile

#### 5.5. Turbulence Intensity Profiles

## 6. Discussion

#### 6.1. Frequency Parameters (Spectrum and Coherence)

#### 6.2. Scaling Parameters (Mean Velocity and Standard Deviation Profiles)

#### 6.3. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Derivation of ADCP Misalignment Error

## References

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**Figure 1.**Simplified illustration of input parameters required to construct flow fields in Tidal Bladed and TurbSim, adapted from TurbSim user’s guide [15].

**Figure 3.**Tidal rose plots showing flow direction as well as ADCP orientation. Solid yellow line corresponds to the instrument heading.

**Figure 4.**Error-ranges for streamwise turbulence intensity calculated from misaligned ADCP data. The ratio ${\sigma}_{v}/{\sigma}_{u}$ was found to be 0.75 at the Sound of Islay tidal site [24]; the ratio $\overline{{u}^{\prime}{v}^{\prime}}/\overline{{u}^{\prime}{}^{2}}$ ranged between 0.3–0.75 at Mahakam River, East Kalimantan, Indonesia [48].

**Figure 5.**Length scales calculated using time-correlation method on rotated instantaneous velocity data. Shaded areas show standard error, which is larger for EMEC-1 due to a shorter dataset.

**Figure 6.**Streamwise length scales calculated by time-correlation method on instantaneous streamwise velocity data. The dashed lines indicate a theoretical streamwise length scale based on open-channel theory, ${L}_{u}=\sqrt{zH}$. The bold dot indicates the point above which the beam spread is larger than the length scales.

**Figure 7.**Comparison of vertical Kaimal and von Kármán models (based on measured length scale) to measured variance normalised spectra for flood cycles. The 10-min spectra are averaged across all flood cycles for velocities 1–3 m/s.

**Figure 8.**Comparison of vertical Kaimal and von Kárm’an models (based on measured length scale) to measured variance normalised spectra for ebb cycles. The 10-min spectra are averaged across all ebb cycles for velocities 1–3 m/s.

**Figure 9.**Comparison of the streamwise Kaimal and Von Kármán models (based on measured length scale) to variance-normalised spectra, measured at EMEC-1 location. The 10-min spectra are averaged across flood and ebb cycles with velocities 1–3 m/s.

**Figure 10.**Comparison of the total Kaimal and von Kármán models (based on measured length scale) to variance-normalised spectra, measured at FORCE-1 location. The 10-min spectra are averaged across flood and ebb cycles for velocities 1–3 m/s.

**Figure 11.**Measured coherence compared to the general IEC coherence model for flood cycles, velocities 1–3 m/s. The shaded area shows the 95% confidence level for measured coherence.

**Figure 12.**Measured coherence compared to the general IEC coherence model for ebb cycles, velocities 1–3 m/s. The shaded area shows the 95% confidence level for measured coherence.

**Figure 13.**Shear profiles for velocities 1–3 m/s. The best-fit power-law exponents and sum squared errors (sse) are displayed for each case.

**Figure 14.**Total turbulence intensity, TKE and anisotropy ratios for all measurement sites for flood (

**top row**) and ebb (

**bottom row**) tides with flows 1–3 m/s.

Reference | Instrument | Sample Rate | Bin Size | Measurement Period | Peak Flow | Depth |
---|---|---|---|---|---|---|

EMEC-1 | Nortek Signature 500 | 4 Hz | 1 m | 10 Apr 2020–22 Apr 2020 | 3.8 m/s | 48 m |

EMEC-2 | Sentinel V50 | 1 Hz | 1 m | 8 Nov 2019–19 Dec 2019 | 3.7 m/s | 48 m |

FORCE-1 | Nortek Signature 500 | 2 Hz | 1 m | 27 Jan 2022–2 Apr 2022 | 4.6 m/s | 38 m |

FORCE-2 | Sentinel V100 | 1 Hz | 0.5 m | 29 Jun 2018–29 Aug 2018 | 4.7 m/s | 34 m |

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**MDPI and ACS Style**

Naberezhnykh, A.; Ingram, D.; Ashton, I.; Culina, J.
How Applicable Are Turbulence Assumptions Used in the Tidal Energy Industry? *Energies* **2023**, *16*, 1881.
https://doi.org/10.3390/en16041881

**AMA Style**

Naberezhnykh A, Ingram D, Ashton I, Culina J.
How Applicable Are Turbulence Assumptions Used in the Tidal Energy Industry? *Energies*. 2023; 16(4):1881.
https://doi.org/10.3390/en16041881

**Chicago/Turabian Style**

Naberezhnykh, Alyona, David Ingram, Ian Ashton, and Joel Culina.
2023. "How Applicable Are Turbulence Assumptions Used in the Tidal Energy Industry?" *Energies* 16, no. 4: 1881.
https://doi.org/10.3390/en16041881