# Validation of Actuator Line Modeling and Large Eddy Simulations of Kite-Borne Tidal Stream Turbines against ADCP Observations

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## Abstract

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^{−1}. This could increase the geographical areas suitable for large-scale tidal power arrays. Numerical modeling of the Deep Green was carried out in a previous study using large eddy simulations and the actuator line method. This numerical model is compared with acoustic Doppler current profiler (ADCP) measurements taken in the wake of a DG operating in a tidal flow under similar conditions. To be comparable, and since the ADCP measures current velocities using averages of beam components, the numerical model data were resampled using a virtual ADCP in the domain. The sensitivity of the wake observations to ADCP parameters such as pulse length, bin length, and orientation of the beams is studied using this virtual ADCP. After resampling with this virtual ADCP, the numerical model showed good agreement with the observations. Overall, the LES/ALM model predicted the flow features well compared to the observations, although the turbulence levels were underpredicted for an undisturbed tidal flow and overestimated in the DG wake 70 m downstream. The velocity deficit in the DG wake was weaker in the observations compared to the LES. The ALM/LES modeling of kite-borne tidal stream turbines is suitable for further studies of array optimization and wake propagation, etc.

## 1. Introduction

^{−1}) [5,6], which limits the possible sites of deployment. Deep Green (DG) by Minesto AB, Gothenburg, Sweden is a kite-borne tidal turbine (see Figure 1) that can harness power from a low-velocity tidal current. The system here studied consists of a 12 m span wing that is anchored to the sea floor with a tether, which also transmits the electrical power generated. A nacelle attached to the wing supports an axial flow turbine. The nacelle also encompasses the generator and power electronics. DG has a control system that steers it in a lemniscate ($\infty $) pattern in a direction that is almost perpendicular to the flow (see Figure 1b). In the lemniscate trajectory, the relative flow speed through the turbine reaches up to 5–10 times the mean tidal current velocity enabling high efficiency of the turbine. The lemniscate trajectory here studied has a horizontal width of 64 m in the cross-stream direction and a vertical extent of 22 m. Eventually, these DG kites can be arranged in large arrays for efficient large-scale power generation. The power plant used in this study is the DG500 with a rated power of 500 kW deployed at the test site outside Holyhead on the west coast of Wales, where the depth is 80 m.

## 2. Materials and Methods

#### 2.1. Numerical Model

#### 2.1.1. Large Eddy Simulations (LES)

#### 2.1.2. Actuator Line Method (ALM)

#### 2.1.3. Computational Setup

^{−1}corresponds to one hour before the tidal peak. It is a fully developed turbulent tidal flow obtained using a precursor analysis. In the precursor analysis, cyclic boundaries were used in the streamwise direction with the tidal forcing represented as a simplified sinusoidal equation i.e.,

#### 2.2. ADCP Observations

^{−1}a, using Taylors frozen eddy hypothesis, turbulent structures smaller than 3.3 m are undersampled.

#### 2.3. Virtual ADCP

#### 2.3.1. Physical ADCP Working Principles

^{−1}in water). The factor of 2 in Equation (8) is to denote the distance travelled by the beam to and from the particle [12,13,48]. ADCPs measure velocity in its coordinate system termed the ADCP coordinate system (${x}_{a},{y}_{a},{z}_{a}$), which is aligned with the beams and the velocity ${\mathbf{u}}_{\mathit{a}}=({u}_{a},{v}_{a},{w}_{a})$ (see Figure 4). The beam can only measure the velocity component parallel to the beam; hence, if the beam was sent vertically downward on the current, only the vertical component of the current velocity, ${w}_{a}$, could be measured (see Figure 4a). To measure the streamwise velocity component ${u}_{a}$, a combination of beams angled as in Figure 4b is required such that the measured velocity ${b}_{1}$ and ${b}_{2}$ is the projection of the flow velocity in the beam directions. Thus, ${b}_{1}$ and ${b}_{2}$ consist of both ${u}_{a}$ and ${w}_{a}$ and can be solved together to obtain the individual components assuming a homogenous flow [49]. ${v}_{a}$ can also be deduced the same way by using two beams angled away from each other in the cross-stream direction (see Figure 4c). Hence, to deduce the 3-dimensional current profile, at least 3 beams are needed (under the assumption of homogenous flow). Since ${w}_{a}$ is measured in both beam combinations, it can be used as a redundancy check for the current homogeneity assumption.

#### 2.3.2. Modeling the Virtual ADCP

#### 2.3.3. Sensitivity Study of the Virtual ADCP

#### 2.4. Turbulence Intensity

## 3. Results

#### 3.1. Numerical Model

#### 3.2. ADCP Observations

#### 3.3. Sensitivity Study of the vADCP

#### 3.4. Comparision of Model Results with Observations

#### 3.4.1. Undisturbed Tidal Flow

^{−1}of mean flow; moreover, using the same trajectory and orientation in a lower mean flow can result in force imbalances. The model velocity has the same shear structure as the observed velocity, although the mean flow velocity is somewhat higher in the model. Further, in Figure 13a, the velocity profile in the model is smoother compared to the ADCP data, which show a non-monotonic increase in speed as a function of depth, even after the time averaging. The grey-shaded plots in Figure 13a correspond to 1 standard deviation ($\sigma $) from the time-averaged velocity at each depth cell.

^{−1}, which is ~7% of the mean flow. The observations have a large variance in $\xi $, which could be an indication of strong flow disturbances or turbulent structures with scales shorter than the distance between opposite beams. The instantaneous error velocity, $\xi $, in the observations reaches values greater than 0.5 ms

^{−1}, indicating that the current homogeneity assumption is less valid in these observations. In the model vADCP, the maximum $\xi $ is about half that of the observations. Higher flow fluctuations, instrument noise, and uncertainties could be the cause of the high $\xi $ in the observations.

#### 3.4.2. Flow in the Deep Green Wake

^{−1}) in the observations. The extreme values of $\xi $ in the observations are not uniform, and are more localized in space, indicating that it could be an effect of noise. In the model vADCP, however, lower $\xi $ values are seen compared to the observations, but it is twice as large as the model without the DG. These high values of $\xi $ imply that the DG wake is highly inhomogeneous in the horizontal direction.

^{−1}for the observations indicating that the current is not homogenous when the DG is operational. In the velocity deficit cores, the $\overline{\xi}$ is also higher because of the periodic flow fluctuations caused by the DG passing through the plane.

## 4. Discussion

^{−1}over a significant period of the tidal cycle, indicating that the DG can be operated effectively in these periods (see Figure 10).

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Symbols | |

${l}_{bin}$ | ADCP cell size [m] |

${l}_{pulse}$ | ADCP pulse length [m] |

$\mathbf{u}=(u,v,w)$ | Velocities in the model coordinate system in $x$, $y,$ and $z$ directions [ms^{−1}] |

${\mathbf{u}}_{a}=({u}_{a},{v}_{a},{w}_{a})$ | Velocities in the ADCP coordinate system [ms^{−1}] |

$\mathbf{x}=(x,y,z)$ | Numerical model coordinate system |

${\mathbf{x}}_{a}=({x}_{a},{y}_{a},{z}_{a})$ | ADCP coordinate system |

$\sigma $ | Standard deviation |

$\xi $ | ADCP error velocity [ms^{−1}] |

$\psi $ | ADCP mount angle: orientation of beam with respect to the flow [deg] |

Abbreviations | |

ADCP | Acoustic Doppler Current Profiler |

ALM | Actuator Line Model |

DG | Deep Green |

LES | Large Eddy Simulations |

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**Figure 1.**(

**a**) Deep Green kite and its parts (

**b**) and the operation of Deep Green in a tidal flow (Minesto AB).

**Figure 2.**(

**a**) Line drawing of the DG element showing the direction vectors of velocity ${U}_{r,i}$, lift, and drag. (

**b**) DG wing with the span-wise actuator line elements sketched with different colors (reworked from Fredriksson et al. [33]).

**Figure 3.**The computational domain used in the numerical model with the Deep Green trajectory and the coordinate system shown.

**Figure 4.**ADCP measured velocity and its components for (

**a**) a single vertical beam; (

**b**) a combination of two beams to compute stream and vertical components, ${u}_{a}$ and ${w}_{a}$; (

**c**) a combination of beams for measuring horizontal and vertical components, ${v}_{a}$ and ${w}_{a}$; and (

**d**) the orientation of beams from the top view to obtain the 3-dimensional current velocity in the ADCP coordinates.

**Figure 5.**(

**a**) Beam geometry showing a grid point $m$ outside the beam and the unit vectors ${\mathbf{e}}_{b}$ and ${\mathbf{e}}_{m}$, where ${J}_{t}$ is the center of the beam transducer. (

**b**) Filtered grid points inside the beam sampling space, filtered using Equation (10) for the four beams in the model coordinate system. B1, B2, B3, and B4 represent the four Janus-configured beams used for estimating the 3-dimensional current profile.

**Figure 6.**Position and orientation of the vADCP with respect to the tidal flow and the DG. $\left(x,y,z\right)$ is the model coordinate system and $({x}_{a},{y}_{a},{z}_{a})$ is the ADCP coordinate system. The orientation of the beam with the flow is defined using the mount angle, $\psi $.

**Figure 7.**Numerical model results with the DG showing the magnitude of the vorticity (equal to 0.25) as a grey isosurface and DG as a blue isosurface.

**Figure 8.**$\overline{u}$ from the model plotted in the $yz$ plane at $x=135\mathrm{m}$, 5 m behind the location of DG (

**a**) without the DG and (

**b**) with DG. The black dashed lines indicate the position $y=15\mathrm{m}$, where the DG trajectory is the widest in $z$.

**Figure 9.**$\overline{u}$ from the model with the DG (

**a**) in the $xy$ plane at $z=44\mathrm{m}$ (center of the lemniscate) and (

**b**) in the $xz$ plane at $y=15\mathrm{m}$ (black dashed line in Figure 8b). (

**c**) The velocity profile of $\overline{u}$ at $\left(x,y\right)=\left(135,15\right)\mathrm{m}$ (red dashed line in Figure 9b) for the undisturbed tidal flow and the flow with the DG.

**Figure 10.**Observed stream velocity (${u}_{o}$) contours for the full observation. The velocity reduction caused due to the DG operation is highlighted using a white square in the figure.

**Figure 11.**Sensitivity study of the (

**a**) time-averaged stream velocity, ${\overline{u}}_{r}$ and the (

**b**) stream component of the turbulence intensity, ${TI}_{x}$, in the undisturbed tidal flow using the vADCP.

**Figure 12.**Sensitivity study of the (

**a**) time-averaged stream velocity, ${\overline{u}}_{r}$, and the (

**b**) stream component of the turbulence intensity, ${TI}_{x}$, in the DG wake using the vADCP at 70 m downstream of the DG.

**Figure 13.**(

**a**) Time-averaged stream velocity from the ADCP observations and the model vADCP in an undisturbed tidal flow; the grey-shaded region corresponds to the standard deviation −$\sigma $ and +$\sigma $ of observed velocity. (

**b**) Time-averaged error velocity, $\overline{\xi}$, for the model vADCP and the observations in an undisturbed tidal flow; the grey-shaded region corresponds to the standard deviation −$\sigma $ and +$\sigma $ of the observed $\overline{\xi}$. $\overline{\xi}$ can be a measure of current homogeneity; the lower the $\overline{\xi}$ value, the more homogenous current in the horizontal direction.

**Figure 14.**Turbulence intensity in the stream direction for the observations and the model vADCP in an undisturbed tidal flow: velocity observations outside three standard deviations of the mean flow velocity are omitted from the turbulence intensity calculations in the observations.

**Figure 15.**Instantaneous velocity showing the cyclic pattern of the DG wake in (

**a**) the observations and (

**b**) the model vADCP 70 m downstream of the DG. Each value in the horizontal axis corresponds to an Eulerian measurement at an interval of 2.2 s (0.455 Hz), and the vertical axis is the depth.

**Figure 16.**(

**a**) Time-averaged stream velocity from the ADCP observations and the model vADCP in the DG wake (70 m downstream); the grey-shaded region corresponds to the standard deviation −$\sigma $ and +$\sigma $ of observed velocity. (

**b**) Time-averaged error velocity, $\overline{\xi}$, for the model and the observations in the DG wake; the grey-shaded region corresponds to the standard deviation −$\sigma $ and +$\sigma $ of the observed $\overline{\xi}$. $\overline{\xi}$ can be a measure of current homogeneity; the lower the $\overline{\xi}$ value, the more homogenous the current in the horizontal layer.

**Figure 17.**Turbulence intensity in the stream direction for the observations and the model vADCP in the DG wake (70 m downstream). Velocities outside three standard deviations of the mean flow velocity are omitted from the turbulence intensity calculations for the observations.

**Figure 18.**Time-averaged stream velocity (

**a**) from the model vADCP and (

**b**) the ADCP observations for undisturbed tidal flow and flow with the DG. (

**c**) Model and observational velocity with DG, normalized with the undisturbed velocity (${\overline{u}}^{DG}\left(z\right)/{\overline{u}}^{UG}\left(z\right)$). The velocities are measured at 70 m downstream of the Deep Green.

**Figure 19.**Turbulence intensity in the stream direction, ${TI}_{x}$, (

**a**) from the model vADCP and (

**b**) the ADCP observations for undisturbed tidal flow and flow with the DG. (

**c**) Model and observational ${TI}_{x}$ with DG, normalized with the undisturbed ${TI}_{x}$ (${{TI}_{x}}^{DG}\left(z\right)/{{TI}_{x}}^{UG}\left(z\right)$): ${TI}_{x}$, is measured at 70 m downstream of the Deep Green.

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

Transducer width, ${l}_{t}$ [m] * | 0.09 |

Beam width, $\alpha $ [deg] * | 3.7 |

Elevation $\beta $[deg]—with respect to the water surface * | −70 |

Direction vector of the beams in the local coordinates * | [−0.2418, −0.2418, −0.9397] |

[0.2418, 0.2418, −0.9397] | |

[−0.2418, 0.2418, −0.9397] | |

[0.2418, −0.2418, −0.9397] | |

Pulse length ${l}_{pulse}$ [m] | 4, 8, 16 |

Bin size ${l}_{bin}$ [m] | 4, 8 |

Orientation with respect to the flow, $\psi $ [deg] | 0, 45 |

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**MDPI and ACS Style**

Prabahar, N.S.S.; Fredriksson, S.T.; Broström, G.; Bergqvist, B.
Validation of Actuator Line Modeling and Large Eddy Simulations of Kite-Borne Tidal Stream Turbines against ADCP Observations. *Energies* **2023**, *16*, 6040.
https://doi.org/10.3390/en16166040

**AMA Style**

Prabahar NSS, Fredriksson ST, Broström G, Bergqvist B.
Validation of Actuator Line Modeling and Large Eddy Simulations of Kite-Borne Tidal Stream Turbines against ADCP Observations. *Energies*. 2023; 16(16):6040.
https://doi.org/10.3390/en16166040

**Chicago/Turabian Style**

Prabahar, Nimal Sudhan Saravana, Sam T. Fredriksson, Göran Broström, and Björn Bergqvist.
2023. "Validation of Actuator Line Modeling and Large Eddy Simulations of Kite-Borne Tidal Stream Turbines against ADCP Observations" *Energies* 16, no. 16: 6040.
https://doi.org/10.3390/en16166040